| L(s) = 1 | − 3-s − 3.70·5-s + 1.70·7-s + 9-s − 1.70·11-s + 6·13-s + 3.70·15-s + 3.70·17-s + 19-s − 1.70·21-s + 4·23-s + 8.70·25-s − 27-s + 2·29-s − 3.40·31-s + 1.70·33-s − 6.29·35-s + 9.40·37-s − 6·39-s + 9.40·41-s − 9.10·43-s − 3.70·45-s + 5.70·47-s − 4.10·49-s − 3.70·51-s − 6·53-s + 6.29·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.65·5-s + 0.643·7-s + 0.333·9-s − 0.513·11-s + 1.66·13-s + 0.955·15-s + 0.897·17-s + 0.229·19-s − 0.371·21-s + 0.834·23-s + 1.74·25-s − 0.192·27-s + 0.371·29-s − 0.611·31-s + 0.296·33-s − 1.06·35-s + 1.54·37-s − 0.960·39-s + 1.46·41-s − 1.38·43-s − 0.551·45-s + 0.831·47-s − 0.586·49-s − 0.518·51-s − 0.824·53-s + 0.849·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9698775479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9698775479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 - 9.40T + 41T^{2} \) |
| 43 | \( 1 + 9.10T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 0.298T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.13787497189113620739854153186, −10.62674926700982840120407675365, −9.137867102446755229704144247143, −8.040785009452444682885448127838, −7.68002840197876840280959547388, −6.42640867867830247282227735713, −5.24839421827732732084775335874, −4.24008708703530120595334536143, −3.28375007747917555130394519532, −0.997603286050187839755930171501,
0.997603286050187839755930171501, 3.28375007747917555130394519532, 4.24008708703530120595334536143, 5.24839421827732732084775335874, 6.42640867867830247282227735713, 7.68002840197876840280959547388, 8.040785009452444682885448127838, 9.137867102446755229704144247143, 10.62674926700982840120407675365, 11.13787497189113620739854153186
